Asymptotic quadratic convergence of the Gauss-Newton method for complex phase retrieval

TL;DR

This paper develops a modified Gauss-Newton method for complex phase retrieval, proving its asymptotic quadratic convergence with high probability under certain measurement conditions.

Contribution

It introduces a novel Gauss-Newton step for complex signals and proves its quadratic convergence without sample splitting.

Findings

Convergence rate is quadratic asymptotically.

Requires $m \\ge O(n \\log^3 n)$ measurements.

Method remains within a region of incoherence during iterations.

Abstract

In this paper, we introduce a Gauss-Newton method for solving the complex phase retrieval problem. In contrast to the real-valued setting, the Gauss-Newton matrix for complex-valued signals is rank-deficient and, thus, non-invertible. To address this, we utilize a Gauss-Newton step that moves orthogonally to certain trivial directions. We establish that this modified Gauss-Newton step has a closed-form solution, which corresponds precisely to the minimal-norm solution of the associated least squares problem. Additionally, using the leave-one-out technique, we demonstrate that $m\ge O( n\log^3 n)$ independent complex Gaussian random measurements ensures that the entire trajectory of the Gauss-Newton iterations remains confined within a specific region of incoherence and contraction with high probability. This finding allows us to establish the asymptotic quadratic convergence rate of the Gauss-Newton method without the need of sample splitting.